Leonardo Modesto

Research Interests

QUANTUM GRAVITY

My main area of expertise is general relativity and
its possible alternative quantizations. I worked in two different approaches:
loop quantum gravity which is a background independent quantization of
general relativity and perturbative quantum gravity. In particular I have recently
introduced an ultra-violet completion of Einstein gravity with the property
to be super-renormalizable. Within this areas, I obtained a number of results
that I am going to explain in more detail.

Super-renormalizable Quantum Gravity

I have studied perturbatively an extension of the Stelle higher derivative
gravity involving an infinite number of derivative terms.
We know that the usual quadratic action is renormalizable but suffers of the
unitarity problem because of the presence of a ghost in the theory.The new theory
is instead ghost-free since the introduction of two entire functions in the model with
the property do not introduce new poles in the propagator.
The local high derivative theory is recovered expanding the entire functions to the lowest
order in the mass scale of the theory. Any truncation of the entire functions gives rise to
the unitarity violation but if we keep all the infinite series we do not fall into these troubles.
The theory is renormalizable at one loop and finite from two loops on. Since only a finite
number of graphs are divergent then the theory is super-renormalizable. I have analyzed
the fractal properties of the theory at high energy showing a reduction of the spacetime
dimension at short scales. Black hole spherical symmetric solutions are also studied
omitting the high curvature corrections in the equation of motions. The solutions are regular
and the classical singularity is replaced by a ``de Sitter-like core&amp;amp;quot; in r=0. Black holes may
show a ``multi-horizon&amp;amp;quot; structure depending on the value of the mass.

Black holes are one of the more mysterious and interesting object predicted by general relativity and
effectively observed in the universe. The space-time structure of a black hole is one of the most
fascinating things in physics. A big problem of the black hole metrics is the singularity problem. In fact we
know the metric has an essential curvature singularity in r=0. Using ideas and technology coming
from LQG we find a possible solution of the problem in quantum gravity.We are able to show the
regularity of space-time at the quantum level but what we consider even more appealing, including
for people outside of LQG, is the semiclassical analysis recently developed. What is really incredible
is the unexpected structure's richness of the metric. The LQG improved metric is regular in r=0 and
very similar to the Reissner-Nordstrom metric showing two event horizons. Looking at the maximal
extension of the spacetime we see that it interpolates between two asymptotically flat regions, the
region at infinity and the r -- 0 region. The metric is form-invariant under the duality symmetry r --a_0/r
(a_0 is proportional to the minimum area in loop quantum gravity and r is the standard Schwarzschild
radial coordinate at asymptotic infinity). This is actually a symmetry between short and large distances.
Of particular interest, the symmetry implies that if an observer in the region at infinity sees a black hole
of mass m an observer in the other asymptotic infinity beyond the horizon (at r -- 0) sees a dual mass
m_P^2/m. A second possibility is an isometric space-time for which the duality extends to an isometry
of the metric. If we consider a semi-polymerization of the metric the result is again a regular singularity
free space-time with two event horizons one r = 2m but the second one now in r=0. We have also studied
the thermodynamics of the semiclassical solutions: temperature, entropy and the evaporation process.
We examined the new loop quantum black hole solution for phenomenological implications and what we
found surprising. We showed that small LQBHs are stable and could be a component of dark matter.
Ultra-light LQBHs created shortly after the Big Bang would now have a mass of approximately 10^{-5} m_P
and emit radiation with a typical energy of about 10^{13} - 10^{14} eV but they would also emit cosmic rays
of much higher energies, albeit few of them. If these small LQBHs form most of the dark matter of the Milky
Way's Halo, the production rate of ultra-high-energy-cosmic-rays (UHECR) by these ultra light black holes
would be compatible with the observed rate of the Auger detector. We want to emphasize that our result is
falsifiable. The theoretical part of the project was developed by myself and developed in collaboration with
Sabine Hossenfelder and Isabeau Pr\'emont-Schwarz. A recent result is related to the stability properties
of the Cauchy horizon. For the first metric the Cauchy horizon is stable for supermassive black holes only
if the polymeric parameter is sufficiently small. For small black holes, however the stability is easily implemented.
The second metric analyzed is not only self-dual but also form-invariant under the transformation r -- r_*^2/r
and r_* = 2 m P. We find that this symmetry protects the Cauchy horizon for any value of the polymeric parameter.
This result was obtained in collaboration with Eric Brown and Robert Mann.

For the future the first goal will be to improve the phenomenological analysis. In particular I want to extract the
emission spectra to compare the prediction with the observational data (this work is related to a collaboration
with the Auger group). Next I will study the compatibility of the new dark matter candidate with the structure
formation in the universe. Direct evidences for primordial quantum black holes will be also considered. After
an improvement of the phenomenological analysis and then I will export what we learned from LBHs to the full
LQG theory. This part of the project is related with the thermodynamical nature of the quantum spacetime.
In particular all the quantities we need are thermodynamical quantities which characterize primordial quantum
black hole and then dark matter and/or UHECRs.

Other future projects related to quantum black holes are :

1) Polymeric Black Holes and Extra Dimensions.

What we learned from minisuperspace reduced models inspired by LQG can be now exported to
any space-time dimension. A very simple extension of our result to extra dimensional scenarios
is to consider a polymeric quantization released by use the Ashtekar variables. The goal is to find
regular and self-dual solutions in any space-time dimension. This research is motivated by the
possibility of decreasing the Planck energy scale in a universe with large extra dimensions.

2) Polymeric Black Holes at LHC.

Once we have a general polymeric black hole solution in any dimension, it will be natural to look for
observable evidences in cosmology, astrophysics and also at the new LHC collider. In particular a
production rate of polymeric black holes will be of phenomenological interest in particle physics.

3) Spinning Loop Black Holes.

We are working now on black holes with angular momentum. Using what we learned in the reduced models
we are able to obtain the semiclassical Kerr black hole from the the Schwarzschild one. The result is already
obtained in two different theories: loop black holes and non-commutative black holes. The space-time structure
is extremely rich and the classical ring singularity disappearances. For the future we want to focus on the
astrophysical implication of modified and/or singularity free spinning black holes.

4) An effective theory for loop black holes.

Using the LQG improved black hole solutions we ask what this a possible effective action for such solutions.
The answer is, for sure, not simple and one could think, as usually happens, that the theory underling loop
black holes is a general action witch contains higher curvature operators. This is a possibility but if we look
to our loop black hole solution we see a strong similarity with the Reissner-Nordstrom metric.
The analogy suggests that a viable theory will involve not curvature corrections to the Einstein-Hilbert action
but an effective non linear action for Einstein-Maxmell non linear electrodynamics, where the polymeric
parameter plays the rule of an effective charge.

Fractal properties of the space-time

For different approaches to quantum gravity a spectral dimension analysis of the space-time at short distances
was been performed. The idea is to consider a diffusion process of classical scalar field on the space-time in a
fictitious time T Using the heat kernel equation in the quantum gravity framework a scale dependent
dimension is obtained. In different theories, causal dynamical triangulations, asymptotic safe quantum gravity,
string theory, ... , the result is always the same: the fractal dimension at hight energy is 2. Actually this is not only
a quantum result because also at the classical level the space-time effective dimension (not the spectral dimension)
reduces to d=2 in the strong gravity regime near a singular point. This result seems to have a universal character
and we think really important to study the same object in LQG or spin-foams (SFs) models. We performed the
calculation of the spectral dimension in 3d and 4d quantum gravity using the dynamics of the Ponzano-Regge
or Turaev-Viro (in 3d) and Barret-Crane (in 4d). We considered a very simple decomposition of the 3d or 4d space-time
already used in the graviton propagator calculation and a boundary state which selects a classical geometry on the
boundary. The results in 3d and 4d are the following. In 3d for the Ponzano-Regge model the spectral dimension of
the space-time runs from 2 to 3, across a 1.5 phase, when the energy of a probe scalar field decreases from high to
low energy; for the Turaev-Viro model the spectral dimension at high energy increases with the value of the cosmological
constant lambda but at low energy the presence of lambda does not change the spectral dimension. For the Barret-Crane
model the spectral dimension of the space-time runs from 2 to 3 but the spectral dimension at the Planck scale depends
on the areas spectrum used in the calculation. For three different are spectrums we find respectively dimension
2.52, 2.62 and 2.12. To improve our preliminary analysis is necessary to introduce a rigorous definition of the heat kernel
equation in LQG and spinfoams. Once we know the heat kernel, the semiclassical regime analysis of the theory is a
straightforward consequence.

In a recent paper we calculated in a different and effective way the spectral dimension of the quantum spacetime.
We implemented the minimal area by averaging the graininess of the quantum manifold in the flat space case. As
a result we obtained that for large diffusion times, the quantum spacetime behaves like a smooth differential manifold
of discrete dimension. On the other hand, for smaller diffusion times, the spacetime looks like a fractal surface
with a reduced effective dimension. For the specific case in which the diffusion time has the size of the minimal area,
the spacetime turns out to have a spectral dimension equal to two, suggesting a possible renormalizable character of
gravity in this regime. For smaller diffusion times, the spectral dimension approaches zero, making less reliable any
physical interpretation in this extreme regime. We extended our result in the presence of a background field and
curvature. We showed that in this case the spectral dimension has a more complicated relation with the diffution time,
making more difficult any conclusion about the renormalizable character of gravity with respect to what found with the
flat space analysis.

Quantum space-time thermodynamics

There must be a deep connection between thermodynamics and general relativity. Since the remarkable discovery
that black holes posses entropy, this connection between Einstein general relativity and thermodynamics has been
slowly unveiled by the work of the likes of Jacobson, Wald, Padnamabhan, ... . In particular Wald found a relation
between the second law of black holes and the diffeomorphisms charge for a generic diff-invariant Lagrangian.
Assuming, as a starting point, holography, Jacobson found that the Einstein equations have a thermodynamical origin.

In a recent paper we used this geometric formalism to read back from a given generic black hole entropy an effective
Lagrangian theory. In other words we applied contrariwise the procedure in the literature: we started from the entropy
and we obtained an effective bulk action. Once we will be able to make this procedure rigorous it will be useful in any
theory where the semiclassical limit is difficult to obtain. In this picture, the fundamental object is no more the action
but the black hole structure near the horizon and its thermodynamic properties.

A similar idea was recently introduced by E. Verlinde which conjectured that gravity is an entropic force. In this framework
we presented a consistency check of the model. We considered at least one well-motivated correction to the area-entropy
relation, the log correction, and following the Verlinde construction, we derived the first correction to Newton's law of gravitation.
We showed that the deviations from Newton's law stemming from the log correction have the same
form as the lowest order quantum effects of perturbative quantum gravity. We consider LQG a natural framework where to
study the thermodynamical properties of the spacetime and we hope to be able in short to reconstruct the semiclassical limit
from the thermodynamical properties of the theory.

Positions Held

February 2004 - December 2005: Post-Doc position at the "Centre di Physique Th\'eorique" of the
``Centre National de la Recherche (C.N.R.S)'', Marseille, France, supported by ``Fondazione Angelo della Riccia''.
February 2004 - August 2004: Post-Doc fellowship at the Physics Department, Turin University, Italy.

November 2000 - November 2003: Graduate studies in Theoretical Physics, University of Torino, Italy;
thesis topic: Born Infeld action, Dp-branes and k-supersymmetry in a new first order formalism;
Advisor: Prof. Pietro Fre',
Area of Study: String Theory.